Linear Algebra Questions

Suppose A is an 5×5 matrix. If rankA=4 , then the columns of A form a basis of R^5. (Yes/No/Maybe)

The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A.

The cofactor expansion of det A along the first row of A is equal to the cofactor expansion of det A along any other row.

The (i,j) minor of a matrix A is the matrix Aij obtained by deleting row i and column j from A.

The determinant of a triangular matrix is the sum of the entries of the main diagonal.

Suppose A is an 5×5 matrix. If A has two pivots, then the dimension of NulA is 2. (Yes/No/Maybe)

If det A is zero, then two columns of A must be the same, or all of the elements in a row or column of A are zero.

A determinant of an n×n matrix can be defined as a sum of determinants of (n−1)×(n−1) submatrices.

Suppose A is an 5×5 matrix. If Ax=0 has only the trivial solution, then ColA=R^5. (Yes/No/Maybe)

If the equation Ax=0 has the trivial solution, then the columns of A span Rn. (Yes/No/Maybe)

Let V be the subset of R^3 consisting of the vectors ⎡⎣a b c⎤⎦ with abc=0. V is closed under scalar multiplication, meaning that if u is in V and c is a real number then cu is in V.

The solution set of the linear system whose augmented matrix [a1a2a3|b] is the same as the solution set of the equation x1a1+x2a2+a3x3=b.

The solution set of a linear system whose augmented matrix is [ a1 a2 a3 |b ] is the same as the solution set of Ax=b, if A=[ a1 a2 a3 ].

The product of any two invertible matrices is invertible. (Yes/No/Maybe)

If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot points. (Yes/No/Maybe)

If the linear transformation TA(x)=Ax is onto, then it is also one-to-one. (Yes/No/Maybe)

The columns of an invertible n×n matrix form a basis for R^n.

If the transpose of A is not invertible, then A is also not invertible. (Yes/No/Maybe)

The null space of an m×n matrix is a subspace of R^m.

The column space of an m×n matrix is a subspace of R^m.